The Great Depression became a global depression for two main reasons. First, there was the interconnectedness (even then) of the world’s economy, which could be harmed by a depression in a major economy like that of the US. Second, there were the effects of some of the policies taken to try to stop the Depression.
First, while the world’s economy was not globalized to the same extent that ours is now, it was fairly connected. There was a great deal of international trade and people in one country would also invest in other countries. Also, the health of the German economy was tied closely to loans from the United States since Germany was still burdened by the reparations it was forced to pay because of the terms of the Treaty of Versailles. When the US economy collapsed, then, other countries were affected. As some examples, Germany stopped being able to borrow from the US, British investors lost money on their US investments, and American investment in Mexico slowed dramatically. Thus, these other three countries were harmed by the Depression in the US.
Second, many countries started to erect trade barriers. In the US, this took the form of the Smoot-Hawley Tariff. These tariffs stifled international trade, which was already hurting because of the drop in demand. The trade barriers threw more people out of work, making the Depression even worse.
For these reasons, the Depression became global, though it must be said that not all countries were equally affected.